Answer: $\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$

## Explanation This question requires application of the omitted variables formula in regression analysis. Given the two single-variable regression models: - $\hat{Y} = 0.5767 + 0.5633X_1$ (omitting $X_2$) - $\hat{Y} = 0.6767 - 0.7633X_2$ (omitting $X_1$) And the covariance/variance information: - $\text{Cov}(X_1, X_2) = 0.603$ - $\text{Var}(X_1) = 0.874$ - $\text{Var}(X_2) = 0.75$ ### Step 1: Calculate the omitted variable bias coefficients The omitted variable bias formula states that when $X_2$ is omitted from the regression: $$\hat{\beta}_1^{simple} = \beta_1 + \beta_2 \cdot \delta_1$$ Where $\delta_1 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}$ Similarly, when $X_1$ is omitted: $$\hat{\beta}_2^{simple} = \beta_2 + \beta_1 \cdot \delta_2$$ Where $\delta_2 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_2)}$ ### Step 2: Calculate the delta values $$\delta_1 = \frac{0.603}{0.874} = 0.6899$$ $$\delta_2 = \frac{0.603}{0.75} = 0.804$$ ### Step 3: Set up the system of equations From the first simple regression: $$0.5633 = \beta_1 + \beta_2 \cdot 0.6899$$ From the second simple regression: $$-0.7633 = \beta_2 + \beta_1 \cdot 0.804$$ ### Step 4: Solve for $\beta_1$ and $\beta_2$ Solving this system: - $\beta_1 = 2.4476$ - $\beta_2 = -2.7312$ ### Step 5: Calculate the intercept The intercept in the multiple regression is: $$\hat{\alpha} = \bar{Y} - \beta_1 \bar{X}_1 - \beta_2 \bar{X}_2$$ From the regression equations, we can derive that: $$\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$$ This matches option D exactly. **Key Insight**: The omitted variable bias formula allows us to recover the true coefficients from simple regressions when we know the covariance structure between the explanatory variables.

Author: Tanishq Prabhu